超级黑箱再探

AI总结：

本文研究了超级黑箱的概念，证明了对于正则的κ和θ存在相应的λ，并探讨了连续着色和自由子集的存在性。

摘要：

设κ,θ<λ为基数，其中λ和κ为正则的。聚焦于一个简单的情况，我们说三元组(λ,κ,θ)具有超级黑箱当以下条件成立。对于某些站稳集S⊆{δ<λ: cf(δ)=κ}和C=⟨Cδ: δ∈S⟩，其中Cδ是δ的闭集，序型为κ，对于每一个着色F=⟨Fδ: δ∈S⟩，其中Fδ: {}^{Cδ}λ→θ，存在⟨cδ: δ∈S⟩∈{}^Sθ使得对于每一个f: λ→θ，对于站稳许多的δ∈S，有Fδ(f|Cδ)=cδ。在先前的工作中，证明了对于一类基数λ，对于许多对(κ,θ)成立。例如，κ<aleph_ω足够大，并且beth_ω(θ)<λ。然而，最有趣的情况（至少就阿贝尔群而言）是κ=aleph_0,aleph_1（这些情况尚未涵盖）。这里我们专注于F为所谓的连续着色的情况，这包括Fδ由某些⟨F_{δ,β}'(f| (Cδ∩β)) : β∈Cδ⟩计算得出。这涵盖了我们所考虑的情况。我们主要证明了没有其他限制的结果：例如，对于每一个正则的κ和θ，存在这样的λ。我们还处理了多个C̄的情况，以及存在相当自由的μ^κ子集的问题。

英文摘要：

Let $ κ, θ< λ$ be cardinals, with $λ$ and $κ$ regular. Concentrating on a simple case, we say that the triple $(λ,κ,θ)$ has a Super Black Box when the following holds. For some stationary $S \subseteq \{δ< λ: cf(δ) = κ\}$ and $\overline C = \langle C_δ: δ\in S \rangle$, where $C_δ$ is a club of $δ$ of order type $κ$, for every coloring $\overline F = \langle F_δ: δ\in S \rangle$ with $F_δ: {}^{C_δ}λ\to θ$, there exists $\langle c_δ: δ\in S\rangle \in {}^S\!θ$ such that for every $f : λ\to θ$, for stationarily many $δ\in S$, we have $F_δ(f \upharpoonright C_δ) = c_δ$. In an earlier work, it was proved (along with much more) that for a class of cardinals $λ$ this holds for many pairs $(κ,θ)$. E.g.~$κ< \aleph_ω$ is large enough, and $\beth_ω(θ) < λ$. However, the most interesting cases (at least with regards to Abelian groups) are $κ= \aleph_0,\aleph_1$ (which have not been covered yet). Here we restrict ourselves to the case where $\overline F$ is a {so-called} \emph{continuous coloring}, which includes the case where $F_δ$ is computed from some $$ \big\langle F_{δ,β}'(f \upharpoonright (C_δ\cap β)) : β\in C_δ\big\rangle. $$ This covers the cases we have in mind. We mainly prove results without any other caveats: e.g. For every regular $κ$ and $θ$ there exists such a $λ$. We also deal with having multiple {$\bar C$-s}, and the existence of quite free subsets of ${}^κμ$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。